Problem: For what positive value of $t$ is $|{-4+ti}| = 2\sqrt{13}$?
Since $|{-4+ti}| = \sqrt{(-4)^2 + t^2} = \sqrt{t^2+16}$, the equation $|{-4+ti}| = 2\sqrt{13}$ tells us that $\sqrt{t^2 + 16} = 2\sqrt{13}$.  Squaring both sides gives $t^2 + 16= 52$, so $t^2= 36$.  Since we want the positive value of $t$, we have $t = \boxed{6}$.